II. Non-Linear Interacting Dark Energy: Analytical Solutions and Theoretical Pathologies

TL;DR

This work extends the study of interacting dark energy to non-linear kernels $Q_{1,2,3}$, deriving dynamical-system viability criteria, exact analytic densities for $Q_{2}$ and $Q_{3}$, and a general reconstruction $\tilde{w}(z)$ for IDE models. The analysis reveals that $Q_{1}$ typically preserves positive energy densities and drives big rip in phantom regimes, while $Q_{2}$ and $Q_{3}$ can induce negative densities depending on energy flow, with $Q_{3}$ offering a potential path to avoid a future big rip if energy transfers from DE to DM. Exact background solutions and the $\tilde{w}(z)$ expressions enable direct comparison of the kernels via statefinder diagnostics, identifying distinctive phantom/quintessence behaviors and CP evolution across epochs. The results provide analytical tools and constraints to guide observational tests of non-linear IDE and broaden the theoretical landscape for dark-sector interactions.

Abstract

We investigate interacting dark energy (IDE) models with phenomenological, non-linear interaction kernels $Q$, specifically $Q_{1}=3Hδ\left(\frac{ρ_{\rm dm}ρ_{\rm de}}{ρ_{\rm dm}+ρ_{\rm de}}\right)$, $Q_{2}=3Hδ\left(\frac{ρ_{\rm dm}^2}{ρ_{\rm dm}+ρ_{\rm de}}\right)$, and $Q_{3}=3Hδ\left(\frac{ρ_{\rm de}^2}{ρ_{\rm dm}+ρ_{\rm de}}\right)$. Using dynamical system techniques developed in our companion paper on linear kernels, we derive new conditions that ensure positive and well-defined energy densities, as well as criteria to avoid future big rip singularities. We find that for $Q_{1}$, all densities remain positive, while for $Q_{2}$ and $Q_{3}$ negative values of either DM or DE are unavoidable if energy flows from DM to DE. We also show that for $Q_{1}$ and $Q_{2}$ a big rip singularity always arises in the phantom regime $w<-1$, whereas for $Q_{3}$ this fate may be avoided if energy flows from DE to DM. In addition, we provide new exact analytical solutions for $ρ_{\rm dm}$ and $ρ_{\rm de}$ in the cases of $Q_{2}$ and $Q_{3}$, and obtain new expressions for the effective equations of state of DM, DE, the total fluid, and the reconstructed dynamical DE equation of state ($w_{\rm dm}^{\rm eff}$, $w_{\rm de}^{\rm eff}$, $w_{\rm tot}^{\rm eff}$, and $\tilde{w}$). Using these results, we discuss phantom crossings, evaluate how each kernel addresses the coincidence problem, and apply statefinder diagnostics to compare the models. These findings extend the theoretical understanding of non-linear IDE models and provide analytical tools for future observational constraints.

Figures (19)

• Figure 1: Dimensionless interaction $Q$ versus redshift for three non-linear interaction models, illustrating when the interaction becomes dominant.
• Figure 2: 3D phase portraits in the iDEDM (left panel, $\delta=+0.1$) and iDMDE (right panel, $\delta=-0.1$) regimes, showing positive energy trajectories at all times for $Q=3H \delta \left(\frac{\rho_{\rm{dm}}\rho_{\rm{de}}}{\rho_{\rm{dm}}+\rho_{\rm{de}}} \right)$.
• Figure 3: 3D phase portraits for $Q=3H \delta \left(\frac{\rho^2_{\rm{dm}}}{\rho_{\rm{dm}}+\rho_{\rm{de}}} \right)$, showing positive-energy trajectories in the iDEDM regime ($\delta=+0.1$, left panel) and negative DE trajectories in the iDMDE regime ($\delta=-0.1$, right panel).
• Figure 4: 2D projection of the phase portraits for $Q=3H \delta \left(\frac{\rho^2_{\rm{dm}}}{\rho_{\rm{dm}}+\rho_{\rm{de}}} \right)$, showing positive-energy trajectories in the iDEDM regime ($\delta=+0.1$, left panel) and negative DE trajectories in the iDMDE regime ($\delta=-0.1$, right panel).
• Figure 5: 3D phase portraits for $Q=3H \delta \left(\frac{\rho^2_{\rm{de}}}{\rho_{\rm{dm}}+\rho_{\rm{de}}} \right)$, showing positive-energy trajectories in the iDEDM regime ($\delta=+0.1$, left panel) and negative DM trajectories in the iDMDE regime ($\delta=-0.1$, right panel).